Theorem breq12 3970

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 3968	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 3969	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 458	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   class class class wbr 3965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966

This theorem is referenced by:  breq12i  3974  breq12d  3978  breqan12d  3981  posng  4655  isopolem  5767  poxp  6173  rbropapd  6183  ecopover  6571  ecopoverg  6574  ltdcnq  7300  recexpr  7541  ltresr  7742  reapval  8434  ltxr  9664  xrltnr  9668  xrltnsym  9682  xrlttr  9684  xrltso  9685  xrlttri3  9686  xposdif  9768  exmidsbthrlem  13555